News From The Front, III

By cjohnson | November 4, 2005 3:35 pm

[Warning! This is an unusually technical post.]

Ok, so last time, I told you a bit about the motivations for what I’ve been up to. Now I want to simply show you some of the product. I’m going to use pictures, words, and equations. I will lose some of you, and for that I’m sorry. But I hope that the words will still give you the gist of the thing. I’ll answer some of your questions in the comments.

Consider the following equation (first found and studied in this context in about 1991/1992 and reported e.g. here, and here, and here):

string equation


The leading boundary conditions for the solutions we wish to consider are:

positive BC

This non-linear differential equation actually contains a lot of string theory information, and it is packaged in a way that is just the sort of thing we dream about in several other parts of string- (and M-) theory: It does not refer to strings by worldsheets, or any way that relies on thinking of the string as a string. We’ve learned from this context and several other studies that when you can identify a string unambiguously in your description, it means more often than not that you are stuck in perturbation theory and so missing a huge amount of the story. So what you look for are ways of defining string theories (or whatever they are since they are about more than strings) without starting with strings.

So how do I find strings? Well, the free energy and partition function (i.e. extremely important defining quantities) for the physical model is given by
free energy

We can develop corrections to the leading behaviour above by just iterating. Actually, you can do this yourself…. substitute in u=z + correction, where “correction” is of order nu (nu is the Greek letter that looks like a curly v above), and then, neglecting anything that is higher order than that, you’ll get s simple equation for the correction, which you can solve. Then you can solve for the next order in the same way, and so on…. You can do this separately for either the large positive z or the large negative z regimes.

The result for positive z regime is
positive u expansion
and so integrating twice and dumping the constant (which turns out to be non-universal physics) we get the free energy:
positive F expansion

I’ve written it in terms of the natural dimensionless combination of parameters which keeps showing up at each term:
string coupling
This is the string coupling! In fact, each term is a term in the “world-sheet” expansion of a string theory…. Have a look (for the technically observant, I’ve not put the sphere term, as it turns out to be non-universal in this example):
positive z worldsheets
So these “world-sheets” are two dimensional surfaces that strings sweep out as they move. A particle sweeps out a line as it moves, a string sweeps out a sheet. To use language from field theory, say, these are the “Feynmann diagrams” for the string theory. Quantum mechanics (yes, this is a quantum theory, and nu plays the role of hbar, Planck’s constant) tells us that we must sum over all paths the strings can take (for a given process), and this is what we see here.

brane picture Notice that the innocent-looking parameter Gamma appears in a special way. Every time there is a boundary on the string world-sheet (so it is an “open string”), there is a factor of Gamma. This actually counts the number of a certain type of “D-brane” that is in the background in which the string is moving. (I described a bit about D-branes here. They are places (dynamical objects) on which string endpoints live. See the picture on the right, showing a snapshot of the strings at one instant, so they are not sweeping out sheets.)

Background? Ah, so these simple string theories have a quite simple spacetime (when it can be identified), which is one reason they are called “minimal” strings. but on the other hand it is a complicated background. This is because there is only one continuous dimension in the target space, but the strength of the string coupling varies from point to point. In fact it grows arbitrarily strong as you move to one end. This is in fact the end that the background D-branes (called “ZZ” branes (link) in this context) are located. There is another type of D-brane in these models called “FZZT” branes (link, link) which stretch along the target space. I might talk about those some other time, since their story is a nice one too.

You might ask whether we have to force Gamma to be positive and an integer by hand, since the equation surely does not care about our stringy interpretation. Turns out that it does. Amazingly, the properties of the equation and its solutions are such that Gamma positive and integer are a very special sector, without you having to impose this! This is a cute result of a study we (James Carlisle, cvj, and Jeff Pennington) did around this time last year, and written up here. That Gamma is positive and integer might remind you of something else that can be counted discretely too. That story is really cute too, and I’ll talk about that in a later post, perhaps. (If you can’t wait, you can read ahead about it from the recent paper we posted on the arXiv on Wednesday.)

So this is all rather nice, I hope you agree. We recover a string theory -with Gamma D-branes- in one perturbative regime of the equation…. we keep expanding and get stringy Feynman diagrams at whatever order we like.

But the really great thing is that we’ve got more than just the perturbation theory! We’ve got every thing else. At this point, we leave most of the field of string theory in the dust, because most of what we can do with strings, as I’ve talked about in other posts, is based upon string perturbation theory. We need to know more about strings, and in particular, we need information to all orders in perturbation theory and we need information about stuff that cannot be described in perturbation theory at all.

Well, we have that here, since the point is that the equation has more than a perturbative expansion. It has a unique exact solution. I can plot it for you here:


So you can ask questions about physics not just pertaining to the extreme right of the solution, but right in the middle, if you want to, where the expansion above makes no sense. This is really a fun and exciting thing to be able to do in such a clear and simple way.

Ok, so you might ask. Hmmm, what is that region to the far left? Well away from the other perturbative regime? Well, you can do the same expansion tricks again to get:
negative u expansion
and hence the free energy:
negative F expansion

Turns out that this is a completely closed string theory. There are no open strings at all! Instead, Gamma appears upon the insertion of a “vertex operator” on the worldsheet, corresponding to the string moving in a background “R-R flux”, a sort of background field in the model generated by closed strings. (Turns out you need an even number of such insertions, as shown in this paper, who clarified a number of key aspects of the modern interpretation of these models.) Here is the picture of what the diagrams encoded in the expansion look like.
negative z worldsheets

So what we’ve found is that there is a completely separate regime encoded by the string equation that represents an entirely closed string theory. This is remarkable, I hope you agree, and this is an example of what is called an “open-closed transition”. Such non-perturbative connections between open and closed strings were discovered in this context a long time before the terms “open-closed transition”, or “open-closed duality” was invented, so people in the field point to other more recent examples as the prototype, such as here and even the heterotic/type I example in here. That’s what I get for being several years ahead of my time, I suppose. (See e.g. the papers here, and here, and here.)

You might ask what such a study does for the field. The simple answers are (1) Proof of principle, and (2) Controlled understanding. In other words, (1) there are several hopes expressed about how non-perturbative string theory might look -including whether it exists- and whether several of the exotic properties -such as dualities, etc- that it has can really be captured in a sensible single model. This (and its cousins) is a concrete example. Note also that we can ask physics questions for any value of the string coupling….i.e. it’s not just duality games. (2) See my comments in the previous post.

Well, that’s probably enough to be getting on with for now. More later.


  • Wolfgang

    Some see the whole universe in a single sandcorn, others see it in a single-plaquette large-N model 😎
    I actually like your “toy models” – ooops, wrong word.

  • Clifford

    Hey, ok, I’m not proud….. I’ll admit that I’m just a big kid: I’m in it for the wonderful toys.


    (Or maybe it’s the models. I forget which.)

  • Helge

    Hey Clifford 😉 I’ll have to get technical on you… (Your equation got me curious). If I get it right, you get an ODE for u = u(z) where Gamma and nue are parameters.
    Now to my questions: What role does z and u play? I would suppose u is some sort of amplitude, and z something like time.
    But if I get it right. You get u: IR -> IR which graph is a line. So how do you come to draw these surfaces?
    These tube like things …

  • Clifford


    z is the coefficient of an operator in the theory, and so since you get u from the partition function by two z derivatives, u is a two-point function of that operator.
    (If you know the older literature it is a cousin of the bulk “puncture” operator….which marks points on surfaces as you sum over them…. dual to a cosmological constant operator…..)

    I’m afraid I don’t know what “IR->IR” means. But the graphs can be drawn because once you’ve identified the string coulpling, there is a unique classification of two dimensional surfaces by topology. There is really nothing else you can draw.

    (There is another motivation which you can forget about for the purposes of this discussion if you don’t know the language…. you can derive this model from an explicit realisation as a matrix model…..from t’Hooft’s large N analysis, the matrix model 1/N expansion builds string worldsheets… turns out the parameter nu is sort of like a (renormalized) 1/N)



  • Clifford

    Oh, Oh, “IR->IR” means map from reals to reals. I see. Well, see my answer anyway.

    The point is that these are numbers each each order that u gives you. In traditional string theory (or field theory or quantum mechanics) these are the results of doing a computation whcih involves summing over all the diagrams of a given topology set by the order you’re working at. (The power of g tells you the order.) So the actual diagram in the picture is there to tell you what the class of diagrams is that gave you that number at that order. There’s a infinite number of such diagrams at each order and you integrate over them to get the result. This way of doing things, you don’t have to do that sort of integration…it’s already done for you and sitting encoded nicely as a term in the asymptotic expansion of the ODE… along with a ton of other physics that you would not be able to represent by any order in that expansion in g. Cool, huh?



  • Helge

    Hey Clifford :-) First questions are solved I think. I took a look at the paper, and the expression for R looks much more complicated there.
    It kinda looks like magic to me right now how you get all that stringy stuff out of the equation. But I completly lack training in string theory …
    I skiming through the paper right now. One more thing I just noticed: You use the u as potential (choice of letter is a lot more obvious now).
    But sure it sounds all extremly cool. :-)

  • Clifford

    It is magic to most string theorists…. the few who have noticed that this stuff exists. It should look like magic. It does very clever stuff in a very different way. this makes it important to know about and to learn, but it is not high fashion, so nobody bothers, hardly. But it is well-defined magic, rooted in a number of concrete computations with checks on the construction coming from many places.

    I will explain later about u also being a potential in a simple one dimensional quantum mechanics problem, and how the wavefunctions of that problem tell you more about D-branes.

    Oh, and maybe I’ll tell how all of this is related to water waves in a canal in Glasgow Edinburgh.



  • Kea

    Goodness, Clifford. This is so cool! Once one has KdV heirarchies one can get there from SDYM, as in

    A self-Dual Yang-Mills Heirarchy and its reductions to Integrable Systems in 1+1 and 2+1 dimensions
    M.J. Ablowitz, S. Chakravarty, L.A. Takhtajan
    Commun. Math. Phys. 158 (1993) 289-314

    or even better, one can do a Twistor correspondence, as in

    Twistor correspondences for the soliton hierarchies
    L.J. Mason, G.A.J. Sparling
    J. Geom. and Phys. 8 (1992) 243-271

    I dragged these references out of my filing cabinet after looking at your paper. Must dust them off and read them….. :)

  • Clifford

    Kea… Yes! Yes! We’re on the same page here! Go for it!


  • Michael D

    Thanks for the post Clifford!

    While much of the string theory diagrams/D-branes is above me, as a physics undergrad there are lots of familar terms which encourages me to make sure that I (try) to understand them in my current courses.

    Partition function and free energy showed up in 3rd year Thermal, and pertubation theory shows up everywhere from QM, to a maths course i’m taking on Applied Fluid Dynamics (eg. matched asymptotic expansions) as well as in Electrodynamics course with dispersion relations in dielectrics.

    What astounds me the most as a I move into higher levels of physics/maths is their reliance on previously learned techniques/knowledge. For example, the non-linear ODE you start with up the top is what you hear about in 1st year “These are complicated, so we won’t do them in thise course.” But in order to solve the higher stuff you of course need to get those basics down.

    While it is very tempting to check out the papers as physics ‘study’, best that I look at them after my exams…



  • Clifford

    Michael D,

    One of the authors of the two papers I mention, Jeff, is an undergraduate. He did not even know much QM when he first started talking with me. He took a book and taught himself what he needed in a short time, and picked up a lot of the other techniques as he went along too. It is a nice combination of string theory and quantum mechanics which showcases the fun stuff you can do with the thigns you learn in undergraduate physics at various levels.



  • Helge

    Hey Michael :-) I am in kinda the same situation as you. Except my next exams are kinda in january, here in Austria. So I get time to look at this stuff right now.

    So one more question to Clifford:
    Are all these conclusions about closed, open strings etc coming up from that QM-Model?

    If yes, it seems a little less magic to me, but even more cooler 😉

    And a big thanks to Clifford for answering my questions :-)

  • Clifford

    Hi Helge,

    The QM model controls a huge amount of the physics in this case. Once you know what u is (solution of the first equation I wrote…sort of a “master” equation), then you put it in the QM and read off a lot of interesting things…. The spectrum means something in the string theory, the wavefunctions themselves mean something, etc…. and hbar in the QM directly maps to the string coupling in the string theory….so quantum corrections in the usual unergraduate QM sense are the same quantum corrections generated by a topological expansion in a string theory, with all the fancy language you’ve heard about involving sums over Riemann surfaces, etc…. Its all there.

    Oh, if you ever wondered what supersymetry was, in essence… that’s there too…the quantum mechanics has a supersymmetric structure!


  • Matt B.

    Excellent! It’s like elegant universe, but with equations. Thank you.

  • Plato

    great…..more solitons. I mean…um solutions.

  • Helge

    Hey Clifford 😉
    You are asking for it. So what is the supersymmetry transform of that Hamilitonian? Or is there one?
    Like for Translations in “time” we would have z -> z + a
    Or timereversal 😉 z -> -z
    And yes I have wondered what supersymmetry is.

    Cheers, Helge

  • Clifford

    Helge. Well, later is probably better for this… but the key point is that the potential u for the case of Gamma number of branes and the potential for the case Gamma+1 form a pair of “superpartner” hamiltonians. The transformation between the two is the famous “Backlund” transformation….but there’s a lot more to this….including how to write the whole story so that it has the same algebraic structure as supersymetry algebras in other systems…. where you can discuss supersymmetry breaking, etc…. have a look at the paper. Oh, there is an excellent review by F. Cooper et al, of supersymmtric QM. Search SPIRES for that name and title combination.



  • Helge

    Well a big thanks again to you Clifford :-)

    For everybody else interested. The paper Clifford was refering to, can be found at:
    Enjoy reading.

    My summary of my understanding (which might be wrong) is: Supersymmetry is kinda like added like you add spin. You double the Hilbertspace. However, you also use a modified Hamiltonian.
    How you modify it can be found in that paper. (Page 13).


  • Clifford

    Well, it is much more than that in general……. its most common realisation is in the context of particle physics where it is a symmetry that relates states (e.g. particles) of integer spin (“bosons”) with particles of half integer spin (“fermions”). The first reason this is interesting is because conventional particle physics has a big separation of the roles of these two types of particle…. matter is made of fermions (quarks, electrons, etc) while force is carried by bosons (photons, gluons, etc)… so if Nature once had a phase where it was supersymmetric, then matter and force are united in that phase….. This is but one of the many reasons that supersymmetry is exciting….. Then, as JoAnne has pointed out, the consequences of the relics of supersymmetry can be quite profound, since supersymmetric scenarios predict particles that may show up at the LHC which may be the principal component of Dark Matter……! Supersymmetry may also be tied up with the unification of forces, the origin of mass…so many things…… so it is interesting for lots of reasons…


  • Ambitwistor

    I think it’s fascinating how an entire string theory can be “hidden” within a simple ordinary differential equation. Are there any examples of field theories arising from ODEs in similarly “non-obvious” ways?

  • Clifford

    Ambitwistor… I’d have to think about that……


  • Plato

    A general link for the laypeople :)

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